Diﬀerential Equations and Matrix Algebra II (MA 222), Winter Quarter, 2000—2001
Exam 2 (100 points)
Name
Be neat and give complete answers. No notes, no text, no Maple.
Box
10 pts 1. Sketch the lines x = 0 and y = 0 (and include the correct hash marks which show the
direction of the trajectories as they cross these lines) for the system of diﬀerential equations.
x
y
=
10 pts 2. Given that the eigenstructure of A =
→
→
x?
the general solution of −
x = A−
1 −2
3 3
−3 6
−4 7
x
y
is
3
2
↔ 1,
1
1
↔ 3, what is
18 pts 3. Given the non—linear system of diﬀerential equations
x = −x2 − xy
y = y 2 + 3x − 4
(a) Verify that (1, −1) is a critical point.
(b) Linearize the above non—linear system at (1, −1). Put your answer in matrix/vector form
in the box below.
Answer:
x
y
=
(c) Given that the eigensystem of the matrix in (b) is
1
2
1√
− 12 i 11
3 1 √
↔ − + i 11,
2 2
1
2
1√
+ 12 i 11
3 1 √
↔ − − i 11
2 2
check the appropriate boxes to indicate the stability of the critical point (1, −1).
Asym Stable Stable Unstable Don’t Know
LS
NLS
3 −4
5 −6
18 pts 4. Given that the eigensystem for
is
4
5
↔ −2,
1
1
↔ −1,
(a) Find the general solution to
x
y
=
3 −4
5 −6
x
y
+
11
17
Put your answer in the box below.
General Solution:
x
y
=
gen
(b) What is the critical point for the above linear system?
(c) Check the appropriate box to indicate the stability of the critical point for the above
linear system.
Asym Stable Stable Unstable Don’t Know
LS
10 pts 5. Consider the second order diﬀerential equation
d2 x
dx
+ 2 − 2 cos(x) = 0
2
dt
dt
(a) Circle the correct adjective: The above is a linear or non—linear diﬀerential equation.
(b) Convert this diﬀerential equation to a Þrst order system (put your answer in the box
below).
Answer:
dx1
=
dt
dx2
=
dt
x = f (x, y)
and A is
y = g(x, y)
the matrix obtained in the linearization process. Check the appropriate box which indicates
stability of the critical point (a, b). (LS = linearized system and NLS = non—linear system)
10 pts 6. Suppose that (a, b) is a critical point for some non—linear system
i) Eigenvalues of A are λ = ±3i.
Asym Stable Stable Unstable Don’t Know
LS
NLS
ii) Eigenvalues of A are λ = −3 and λ = 2.
Asym Stable Stable Unstable Don’t Know
LS
NLS
14 pts 7. Each of the direction Þelds on the handout comes from a 2 × 2 linear system of diﬀerential
equations. For each of the following conditions, put the letter (or letters) of the graph(s)
on the handout which is (are) described in the condition. If no graph satiÞes the condition,
simply write NA (not applicable).
a. The critical point is at the origin.
b. The system is non—homogeneous.
c. The critical point is asymptotically stable.
d. One eigenvalue is positive and the other one is negative.
e. The critical point is a stable center (exactly one graph satiÞes this condition).
f. The critical point is not at the origin and is unstable.
g. The matrix has complex eigenvalues.